Chapter 6 Applied Problems: Guided Assignment

Published

October 16, 2026

library(tidyverse)
library(tidymodels)
library(ISLR2)

# step_pls() requires the Bioconductor package mixOmics.
# Check this at the start so Quarto fails early with a useful message.
if (!requireNamespace("mixOmics", quietly = TRUE)) {
  stop(
    "mixOmics is not available to the R session used by Quarto. ",
    "Install it in this same R installation with BiocManager::install('mixOmics'), ",
    "then restart R/RStudio and render again."
  )
}
library(mixOmics)

Introduction

Use the approaches introduced in the Chapter 6 lab to complete the following applied problems. Continue to use tidymodels and the tidyverse. When tuning a model, use cross-validation on the training data and keep the test set untouched until the final model has been selected.

For every tuning procedure:

set a random seed;
use the same resamples when comparing candidate values of a tuning parameter;
inspect the tuning plot rather than relying only on one numerical result;
finalize the selected workflow and fit it to the entire training set;
report test RMSE and test \(R^2\) only after model selection is complete.

Problem 1: Predicting college applications

Problem 1 — Chapter 6, Problem 9 with the assignment modifications

Use the College data to predict the number of applications received, Apps, from the remaining variables.

You will compare:

ordinary least-squares regression,
ridge regression,
lasso regression,
principal components regression (PCR),
partial least squares (PLS).

For the tuned models, use 10-fold cross-validation. For the final comparison, create a properly formatted table with the model name, model complexity, test RMSE, and test \(R^2\).

Part (a): Prepare the data, split, and resamples

Guidance

Convert College to a tibble and examine its variables.
Split the data into training and test sets, using Apps for stratification.
Create 10 folds from the training data for later cross-validation.
Create one common preprocessing recipe that can be reused by all models.
The College data include a categorical predictor, so account for new levels and create dummy variables.
Normalize numeric predictors so that regularized models are not affected by differences in measurement scale.

Useful functions include initial_split(), training(), testing(), vfold_cv(), recipe(), step_novel(), step_dummy(), step_zv(), and step_normalize().

Part (b): Ordinary linear regression

Guidance

Fit an ordinary linear regression model using all remaining variables to predict Apps.

Use a workflow containing the common recipe and a linear_reg() model with the lm engine. Fit the workflow to the full training set and evaluate it on the test set.

Report:

test RMSE;
test \(R^2\);
the number of predictors retained by this model.

The ordinary linear model does not remove predictors, so its complexity count is the number of predictor columns after preprocessing.

Part (c): Ridge regression

Guidance

Fit a ridge regression model using the same predictors and preprocessing.

Use linear_reg() with the glmnet engine.
Set mixture = 0 for ridge regression.
Set penalty = tune().
Create a grid of possible penalty values and tune with the 10 folds created in part (a).
Use autoplot() to examine the tuning results.
Use show_best() to inspect the best candidate values.
Select the best penalty, finalize the workflow, and fit the final model to all training observations.

Report:

selected penalty value;
test RMSE;
test \(R^2\);
model complexity.

Remember that ridge regression shrinks coefficients but generally does not set them exactly to zero.

Part (d): Lasso regression

Guidance

Repeat the regularization analysis using the lasso.

Set mixture = 1.
Tune penalty using the same 10-fold resamples.
Display the tuning results with autoplot().
Finalize and fit the selected model.
Apply tidy() to the final fitted model to inspect its coefficients.

Report:

selected penalty value;
test RMSE;
test \(R^2\);
number of non-zero predictor coefficients retained by the final lasso model.

Do not count the intercept when determining model complexity.

Part (e): Principal components regression (PCR)

Guidance

Fit a PCR model using a PCA preprocessing step followed by ordinary linear regression.

For this assignment, tune the PCA variance threshold using cross-validation.

Start from the common preprocessing recipe.
Add step_pca(all_predictors(), threshold = tune()).
Create a reasonable grid of threshold values.
Tune using the same 10 folds.
Use autoplot() and the tuning results to select the best threshold.
Finalize and fit the model.

Report:

selected threshold;
test RMSE;
test \(R^2\);
number of principal components retained by the final model.

A useful way to count retained components is to examine the final model with tidy() and count the non-intercept component terms.

Part (f): Partial least squares (PLS)

Guidance

Fit a PLS model using the common preprocessing recipe.

Add step_pls(all_predictors(), num_comp = tune(), outcome = "Apps").
Create a grid of possible values for num_comp.
Tune using the same 10 folds.
Display the tuning results with autoplot().
Select the best number of components, finalize the workflow, and fit the final model.

Report:

selected number of components;
test RMSE;
test \(R^2\).

Part (g): Compare the models

Guidance

Create a properly formatted comparison table with one row for each model and these columns:

short model name;
number of predictors/components retained;
test RMSE;
test \(R^2\).

The table should contain the ordinary linear model, ridge, lasso, PCR, and PLS.

Use the table to answer the following questions:

Which model has the smallest test RMSE?
Which model has the largest test \(R^2\)?
Are the performance differences large enough to clearly favor one method?
Which models are simpler in terms of the number of retained predictors/components?
If two models have nearly identical predictive performance, is there a reason to prefer the simpler one?

Remember that ordinary regression and ridge retain all predictor columns, while lasso may eliminate predictors and PCR/PLS reduce dimension through components.

Problem 2: Predicting Boston crime rate

Problem 2 — Chapter 6, Problem 11 with the assignment modifications

Use the Boston data to predict per-capita crime rate, crim.

Fit and tune the following four models:

ridge regression,
lasso regression,
PCR,
PLS.

Use cross-validation to tune each model. Display a tuning plot for each method and create a final comparison table similar to the one from Problem 1.

Part (a): Prepare the data and common resamples

Guidance

Convert Boston to a tibble.
Split the data into training and test sets, stratifying by crim.
Create 10 folds from the training data.
Create common linear-regression and glmnet model specifications.
Create a common recipe that prepares all predictors for the models.

Use the same split and the same folds for every model so the comparisons are fair.

Ridge regression

Guidance

Use mixture = 0 and penalty = tune().
Choose a reasonable penalty grid.
Tune using the 10 folds.
Display the tuning results with autoplot().
Select the best model using RMSE.
Finalize, fit, and evaluate on the test set.

Record the selected penalty, test RMSE, and test \(R^2\).

Lasso regression

Guidance

Use mixture = 1 and penalty = tune().
Tune using 10-fold cross-validation.
Display the tuning results.
Finalize and fit the selected model.
Use tidy() to inspect the final coefficients.

Record:

selected penalty;
test RMSE;
test \(R^2\);
number of non-zero predictor coefficients.

Principal components regression (PCR)

Guidance

For this problem, tune PCR directly by the number of components.

Add step_pca(all_predictors(), num_comp = tune()) to the common recipe.
Tune plausible component counts with the same 10 folds.
Display the tuning plot.
Select the best number of components using RMSE.
Finalize, fit, and evaluate on the test set.

Record the selected number of components, test RMSE, and test \(R^2\).

Partial least squares (PLS)

Guidance

Add step_pls(all_predictors(), num_comp = tune(), outcome = "crim").
Tune the number of components with the same 10 folds.
Display the tuning plot.
Select the best number of components using RMSE.
Finalize, fit, and evaluate on the test set.

Record the selected number of components, test RMSE, and test \(R^2\).

Performance summary

Guidance

Create a table with one row for each of the four methods and the following columns:

Model;
Predictors / Components;
Test RMSE;
Test \(R^2\).

Use the table—not training performance—to compare the models.

Part (b): Choose a best model or set of models

Guidance

Using your comparison table, propose either:

one preferred model, or
a small set of models that appear essentially competitive.

Justify your choice using both predictive performance and model complexity. If several models have very similar RMSE and \(R^2\), say so rather than overstating small differences.

Part (c): Does the chosen model use all predictors?

Guidance

Answer this question in a way appropriate for the model you selected.

For the lasso, inspect which original predictor coefficients are non-zero.
For PCR or PLS, be careful: using fewer components does not necessarily mean that only a few original variables contribute. Each component can be a linear combination of many or all original predictors.

Explain clearly what “uses all predictors” means for your chosen model.

Submission checklist

Before submitting, verify that your report includes:

reproducible random seeds;
one untouched test set for each problem;
10-fold cross-validation for every tuned model;
tuning plots for ridge, lasso, PCR, and PLS;
selected hyperparameter values;
test RMSE and test \(R^2\) for every final model;
appropriate model-complexity counts;
a formatted model-comparison table for each problem;
written conclusions based on the test-set results and model complexity.